Koch snowflake

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The Koch snowflake (also known as the Koch star and Koch island[1]) is a mathematical curve and one of the earliest fractal curves to have been described. It is based on the Koch curve, which appeared in a 1904 paper titled "On a continuous curve without tangents, constructible from elementary geometry" (original French title: "Sur une courbe continue sans tangente, obtenue par une construction géométrique élémentaire") by the Swedish mathematician Helge von Koch.

Contents

Construction

The Koch curve can be constructed by starting with an equilateral triangle, then recursively altering each line segment as follows:

After one iteration of this process, the result is a shape similar to the Star of David.

The Koch curve is the limit approached as the above steps are followed over and over again.

Properties

The Koch curve has an infinite length because each time the steps above are performed on each line segment of the figure there are four times as many line segments, the length of each being one-third the length of the segments in the previous stage. Hence the total length increases by one third and thus the length at step n will be (4/3)n of the original triangle perimeter: the fractal dimension is log 4/log 3 ≈ 1.26, greater than the dimension of a line (1) but less than Peano's space-filling curve (2).

The Koch curve is continuous everywhere but differentiable nowhere.

Taking s as the side length, the original triangle area is \frac{s^2\sqrt{3}}{4} . The side length of each successive small triangle is 1/3 of those in the previous iteration; because the area of the added triangles is proportional to the square of its side length, the area of each triangle added in the nth step is 1/9th of that in the (n-1)th step. In each iteration after the first, 4 times as many triangles are added as in the previous iteration; because the first iteration adds 3 triangles, the nth iteration will add  3 \cdot 4^{n-1} triangles. Combining these two formulae gives the iteration formula:

where A0 is area of the original triangle. Substituting in

and expanding yields:

In the limit, as n goes to infinity, the limit of the sum of the powers of 4/9 is 4/5, so

So the area of a Koch snowflake is 8/5 of the area of the original triangle, or \frac{2s^2\sqrt{3}}{5}.[2] Therefore the infinite perimeter of the Koch triangle encloses a finite area.

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